Aula 23 - Função de 2º Grau - Parte 1- Definição, equação, gráfico e vértice.

Introduction

In this tutorial, we will explore the quadratic function, also known as the function of the second degree. We will understand its geometric properties, derive its formula, and learn how to graph it. Additionally, we will identify the vertex coordinates of the parabola. This knowledge is essential for anyone studying mathematics, particularly in algebra and calculus.

Step 1: Understanding the Quadratic Function

• The general form of a quadratic function is: [ f(x) = ax^2 + bx + c ] where:
  • ( a ) determines the direction of the parabola (upwards if ( a > 0 ), downwards if ( a < 0 )).
  • ( b ) influences the position of the vertex along the x-axis.
  • ( c ) represents the y-intercept.

Practical Advice

• Make sure to identify the values of ( a ), ( b ), and ( c ) for any quadratic function you encounter.
• Remember that the vertex is a critical point that helps in graphing the parabola.

Step 2: Graphing the Quadratic Function

• To graph a quadratic function, follow these steps:
  1. Identify the vertex using the formula: [ x_v = -\frac{b}{2a} ] Substitute ( x_v ) back into the function to find ( y_v ).
  2. Calculate additional points on either side of the vertex by substituting values for ( x ) into the function.
  3. Plot these points on a coordinate system.
  4. Draw a smooth curve through the points to form the parabola.

Practical Advice

• Use symmetry to find points. The parabola is symmetric about the vertex.
• Ensure you label the vertex and other points clearly on your graph.

Step 3: Special Points of the Parabola

• There are four special points to consider when graphing:
  1. The vertex, as discussed.
  2. The y-intercept, found by evaluating ( f(0) = c ).
  3. The x-intercepts, if they exist, are found by solving the equation ( ax^2 + bx + c = 0 ) using the quadratic formula: [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
  4. Other points determined by selecting values of ( x ) around the vertex.

Practical Advice

• Always check the discriminant (( b^2 - 4ac )) to determine the nature of the roots (real and distinct, real and equal, or complex).

Step 4: Finding the Vertex Coordinates

• The vertex coordinates can be summarized as: [ V\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) ]
• This point is crucial for determining the maximum or minimum value of the quadratic function.

Practical Advice

• Use a graphing calculator or software to visualize how the vertex shifts with different values of ( a ), ( b ), and ( c ).

Conclusion

In this tutorial, we covered the fundamental aspects of the quadratic function, including its definition, graphing techniques, and the significance of the vertex. Understanding these concepts is essential for higher-level mathematics. As a next step, practice graphing various quadratic functions and identifying their key features. This will solidify your understanding and prepare you for more advanced topics in algebra and calculus.